🚀 go-pugleaf

Androcles wrote:
> 
> Hana Bizek <hbizek@ameritech.net> wrote in message
> news:387DD6C1.24CC@ameritech.net...
> > Hello, all,
> > some of you  will know me from previous postings. My  quest for
> > 3-dimensional Rubik's cobe designers is still a dismal zero. There is a
> > real danger that I am the first to have come up with this. Please, if
> > you know of anyone who does  something similar to what I have done, let
> > me know. The page to look at is http://cube.misto.cz.
> If duplicating your website information means producing the image of a white
> cat playing with a snowball on ice-covered tundra, then I'm sure I can
> reproduce it. It is totally blank.
> Androcles

Thanks for your post. Of course you can construct a clean design, whose
one side is completely white. But you need to decide if you want this
clean cube to have six colors overall or three colors overall.  If the
latter, you need parity pairs.
Hana a kostky

Even better if I could get it to download or the page was readable without
the background.

Actually.

Look at the back numbers of scientific American. They had various designs of
tetrahedron based etc etc.
Hypercube - that is new I think.

Ishihama Yoshiaki <ishmnn@cap.bekkoame.or.jp> wrote in message
news:ishmnn-1501000106000001@p84933c.mta4.ap.so-net.ne.jp...
> In article <387DD6C1.24CC@ameritech.net>, hbizek@ameritech.net wrote:
>
> > Hello, all,
> > some of you  will know me from previous postings. My  quest for
> > 3-dimensional Rubik's cobe designers is still a dismal zero. There is a
> > real danger that I am the first to have come up with this. Please, if
> > you know of anyone who does  something similar to what I have done, let
> > me know. The page to look at is http://cube.misto.cz. Please go down to
> > the Appendices and click on "Thre-dimensional designs of Dr. Hana M.
> > Bizek." Thank you.
>
> 3D Rubik cube is out of date, too old.   Forget that.
> Check my 4D rubik Cube applet please.
> I also made 5D Rubik Cube simulations for Mac.
>
> http://www.bekkoame.ne.jp/~ishmnn/java/java.html
>
> --
> Ishihama Yoshiaki
> E-mail: ishmnn@cap.bekkoame.ne.jp
> URL: http://www.asahi-net.or.jp/~hq8y-ishm/

Ishihama Yoshiaki wrote:
> 
> In article <387DD6C1.24CC@ameritech.net>, hbizek@ameritech.net wrote:
> 
> > Hello, all,
> > some of you  will know me from previous postings. My  quest for
> > 3-dimensional Rubik's cobe designers is still a dismal zero. There is a
> > real danger that I am the first to have come up with this. Please, if
> > you know of anyone who does  something similar to what I have done, let
> > me know. The page to look at is http://cube.misto.cz. Please go down to
> > the Appendices and click on "Thre-dimensional designs of Dr. Hana M.
> > Bizek." Thank you.
> 
> 3D Rubik cube is out of date, too old.   Forget that.
> Check my 4D rubik Cube applet please.
> I also made 5D Rubik Cube simulations for Mac.
> 
> http://www.bekkoame.ne.jp/~ishmnn/java/java.html
> 
> --
> Ishihama Yoshiaki
> E-mail: ishmnn@cap.bekkoame.ne.jp
> URL: http://www.asahi-net.or.jp/~hq8y-ishm/

I am talking here about designs, not individual Rubik's cubes. But I
will look at yoir page.
Hana a kostky

Hana Bizek <hbizek@ameritech.net> wrote in message
news:387F0091.BEA@ameritech.net...
> Androcles wrote:
> >
> > Hana Bizek <hbizek@ameritech.net> wrote in message
> > news:387DD6C1.24CC@ameritech.net...
> > > Hello, all,
> > > some of you  will know me from previous postings. My  quest for
> > > 3-dimensional Rubik's cobe designers is still a dismal zero. There is
a
> > > real danger that I am the first to have come up with this. Please, if
> > > you know of anyone who does  something similar to what I have done,
let
> > > me know. The page to look at is http://cube.misto.cz.
> > If duplicating your website information means producing the image of a
white
> > cat playing with a snowball on ice-covered tundra, then I'm sure I can
> > reproduce it. It is totally blank.
> > Androcles
>
> Thanks for your post. Of course you can construct a clean design, whose
> one side is completely white. But you need to decide if you want this
> clean cube to have six colors overall or three colors overall.  If the
> latter, you need parity pairs.
> Hana a kostky
Hana, what I'm saying is that when I attempted to visit
http://cube.misto.cz.
I found a blank page :-)
Androcles

On Fri, 14 Jan 2000 21:02:01 GMT, "Androcles" <androcles@home.com>
wrote:
>Hana, what I'm saying is that when I attempted to visit
>http://cube.misto.cz.
>I found a blank page :-)
>Androcles

Try 
http://cube.misto.cz 
without the full stop

Androcles wrote:
> 
> Hana Bizek <hbizek@ameritech.net> wrote in message
> news:387F0091.BEA@ameritech.net...
> > Androcles wrote:
> > >
> > > Hana Bizek <hbizek@ameritech.net> wrote in message
> > > news:387DD6C1.24CC@ameritech.net...
> > > > Hello, all,
> > > > some of you  will know me from previous postings. My  quest for
> > > > 3-dimensional Rubik's cobe designers is still a dismal zero. There is
> a
> > > > real danger that I am the first to have come up with this. Please, if
> > > > you know of anyone who does  something similar to what I have done,
> let
> > > > me know. The page to look at is http://cube.misto.cz.
> > > If duplicating your website information means producing the image of a
> white
> > > cat playing with a snowball on ice-covered tundra, then I'm sure I can
> > > reproduce it. It is totally blank.
> > > Androcles
> >
> > Thanks for your post. Of course you can construct a clean design, whose
> > one side is completely white. But you need to decide if you want this
> > clean cube to have six colors overall or three colors overall.  If the
> > latter, you need parity pairs.
> > Hana a kostky
> Hana, what I'm saying is that when I attempted to visit
> http://cube.misto.cz.
> I found a blank page :-)
> Androcles

Androcles, I clicked the URL given  in your message. First you will see
an outline for algorithms to solve the Rubik's cube. That is the work of
Josef Jelinek, my collaborator. My designs are found in the Appendices.
The page is hardly blank. Please try again.
Hana a kostky
PS Why not get a set of Rubik's cubes and try  to do some designs? 27
cubes will be sufficient for majority of designs in
http://cube.misto.cz.

Hana Bizek <hbizek@ameritech.net> wrote in message
news:387DD6C1.24CC@ameritech.net...
> Hello, all,
> some of you  will know me from previous postings. My  quest for
> 3-dimensional Rubik's cobe designers is still a dismal zero. There is a
> real danger that I am the first to have come up with this. Please, if
> you know of anyone who does  something similar to what I have done, let
> me know. The page to look at is http://cube.misto.cz.
If duplicating your website information means producing the image of a white
cat playing with a snowball on ice-covered tundra, then I'm sure I can
reproduce it. It is totally blank.
Androcles

In article <387DD6C1.24CC@ameritech.net>, hbizek@ameritech.net wrote:

> Hello, all,
> some of you  will know me from previous postings. My  quest for
> 3-dimensional Rubik's cobe designers is still a dismal zero. There is a
> real danger that I am the first to have come up with this. Please, if
> you know of anyone who does  something similar to what I have done, let
> me know. The page to look at is http://cube.misto.cz. Please go down to
> the Appendices and click on "Thre-dimensional designs of Dr. Hana M.
> Bizek." Thank you.

3D Rubik cube is out of date, too old.   Forget that.
Check my 4D rubik Cube applet please.
I also made 5D Rubik Cube simulations for Mac.

http://www.bekkoame.ne.jp/~ishmnn/java/java.html

-- 
Ishihama Yoshiaki
E-mail: ishmnn@cap.bekkoame.ne.jp
URL: http://www.asahi-net.or.jp/~hq8y-ishm/

Sorry, Hana, but I still cannot get anything.
My browser is Internet Explorer 5.0, and
it automatically changes the URL to
 http://cube.misto.cz/
It seems that roger bamkin is having trouble, too.
Androcles

Androcles wrote:
> 
> Sorry, Hana, but I still cannot get anything.
> My browser is Internet Explorer 5.0, and
> it automatically changes the URL to
>  http://cube.misto.cz/
> It seems that roger bamkin is having trouble, too.
> Androcles

My browswr is Netscape and I have no difficulty viewing this file. But
if you have problems, don't worry about them. Thank you fo trying.
Hana a kostky

In rec.puzzles Hana Bizek <hbizek@ameritech.net> wrote:
: Androcles wrote:
:> 
:> Sorry, Hana, but I still cannot get anything.
:> My browser is Internet Explorer 5.0, and
:> it automatically changes the URL to
:>  http://cube.misto.cz/
:> It seems that roger bamkin is having trouble, too.
:> Androcles

: My browswr is Netscape and I have no difficulty viewing this file. But
: if you have problems, don't worry about them. Thank you fo trying.
: Hana a kostky


I tried with both Internet Exploder 5.0 and Netscape 4.6 and even with 
Lynx and it seems to work perfectly in any of those.

Very nice designs you have there, Hana.

<zam> 

Silvennoinen Sami wrote:
> 
> In rec.puzzles Hana Bizek <hbizek@ameritech.net> wrote:
> : Androcles wrote:
> :>
> :> Sorry, Hana, but I still cannot get anything.
> :> My browser is Internet Explorer 5.0, and
> :> it automatically changes the URL to
> :>  http://cube.misto.cz/
> :> It seems that roger bamkin is having trouble, too.
> :> Androcles
> 
> : My browswr is Netscape and I have no difficulty viewing this file. But
> : if you have problems, don't worry about them. Thank you fo trying.
> : Hana a kostky
> 
> I tried with both Internet Exploder 5.0 and Netscape 4.6 and even with
> Lynx and it seems to work perfectly in any of those.
> 
> Very nice designs you have there, Hana.
> 
> <zam>

Thank you. And they can be done because I have done them -all of them -
by comventionally twiddling the cobes. Needless to say, one should be
able to solve the Rubik's cube completely as a prequisite. So far I am
failing in my search for people who can do this type of thing.
Hana a kostky

)Thank you. And they can be done because I have done them -all of them -
)by comventionally twiddling the cobes. Needless to say, one should be
)able to solve the Rubik's cube completely as a prequisite. So far I am
)failing in my search for people who can do this type of thing.
)Hana a kostky

You should also be able to duplicate a cube.  Interesting challenge,
that:  Scramble a cube, give each contestant a solved cube, and let
them recreate the scrambled cube.

Anyhow, I looked at the designs on your page, and I can safely say that
if I ever have the money to blow on it, I'll go out and buy a couple
dozen cubes to make designs with.

I already have an idea for the first design I'll make: A worm, i.e.
a signle line which walks around the entire cube surface.
(Would make an interesting puzzle in itself to fit the cubes together..)
Would be even nicer if the lines also walk around inside surfaces, so
that when you remove a cube, you still have one line..

By the way, do your designs also have an internal structure ??

SaSW,
-- 
Willem (at stack dot nl)
Disclaimer: I am in no way responsible for any of the statements
            made in the above text. For all I know I might be
            drugged or something..
            No I'm not paranoid. You all think I'm paranoid, don't you !
#EOT

Willem-Jan Monsuwe wrote:
> 
> )Thank you. And they can be done because I have done them -all of them -
> )by comventionally twiddling the cobes. Needless to say, one should be
> )able to solve the Rubik's cube completely as a prequisite. So far I am
> )failing in my search for people who can do this type of thing.
> )Hana a kostky
> 
> You should also be able to duplicate a cube.  Interesting challenge,
> that:  Scramble a cube, give each contestant a solved cube, and let
> them recreate the scrambled cube.
>
That brings me to constructing individual patters on a cube. People do
this, and you can find such patterns on the web.  This is the first step
of a simple, non-refined design algorithm. Since I cannot memorize
things well, I have developed an algorithm to meet these needs of as
little rote as possible.
> Anyhow, I looked at the designs on your page, and I can safely say that
> if I ever have the money to blow on it, I'll go out and buy a couple
> dozen cubes to make designs with.
> 
For a minimum design, you need 8 cubes. 27 cubes is better. You can go
higher, but it becomes more expensive.
> I already have an idea for the first design I'll make: A worm, i.e.
> a signle line which walks around the entire cube surface.
> (Would make an interesting puzzle in itself to fit the cubes together..)
The second easy step of the unrefined design algorithm is to do just
that. And yes, it IS a puzzle in itself. I call it the color control.
Color control makes the  design algorithm three times as complex as the
solution algorithm. But it has to be done, otherwise yoou have no
design.
> Would be even nicer if the lines also walk around inside surfaces, so
> that when you remove a cube, you still have one line..

I found out that some of the links in http://cube.misto.cz will give you
error messages. You see, I got these files on the web and they have
migrated. The most serious such migration occurred in a file, where I
have 3 designs. I asked the owner, Jiri Fridrich, to put those on the
web, before I obtained the cz* site. They have changed his server. Look
at http://www.binghamton.edu/fridrich/hana1.html and click on Jarmila
design. It might give you some tips on whaat you want.
> 
> By the way, do your designs also have an internal structure ??
> 
Noi, not offhand. But the designs' interior has not been neglected.You
can create a "design within a design. Or you can set a certain criterion
and investigate the design's sytmmetry. Rhis is one if the subproblems
of the design problem. 
The design priblem has ramifications in fuelds in science and math. I
know it sounds strange, but it spans fractal geometry and parity.
Hana a kostky
> SaSW,
> --
> Willem (at stack dot nl)
> Disclaimer: I am in no way responsible for any of the statements
>             made in the above text. For all I know I might be
>             drugged or something..
>             No I'm not paranoid. You all think I'm paranoid, don't you !
> #EOT

Hana, has it occured to you that Rubik's cube and the rotations form a
group? (I mean in the mathematical sense.)
Androcles


Hana Bizek <hbizek@ameritech.net> wrote in message
news:387DD6C1.24CC@ameritech.net...
> Hello, all,
> some of you  will know me from previous postings. My  quest for
> 3-dimensional Rubik's cobe designers is still a dismal zero. There is a
> real danger that I am the first to have come up with this. Please, if
> you know of anyone who does  something similar to what I have done, let
> me know. The page to look at is http://cube.misto.cz. Please go down to
> the Appendices and click on "Thre-dimensional designs of Dr. Hana M.
> Bizek." Thank you.
>
> Instead of providing me with a name of a fellow designer, the above page
> ended up as a link to rather prestigious sites. Some are:
> * Mensa CR at http://www.mensa.cz/cz/odkazy
> * Sir Roger Penrose Society at http://welcome.to/SRPS
> * The Geometry Junkyard (I forgot the URL, will look it up upon request)
> * MSO Worldwide, Ltd at http://www.msoworld.com
>
> The last site, MSO Worldwide, has launched a new links directory. Our
> site belongs to the Mental Skills - Mechanical Puzzles category. Please
> have a look at it, and, if you think it's good, vote for our site. And
> look around. It's a fascinating site for any and all puzzle lovers, and
> math lovers.
>
> Cheers,
>
> Hana M. Bizek
> hbizek@ameritech.net

Androcles wrote:
> 
> Hana, has it occured to you that Rubik's cube and the rotations form a
> group? (I mean in the mathematical sense.)
> Androcles
> 
Jesus Christ, Androcles!
I always knew that all elements (or states) ofd a Rubik's cube, or any
other cubes, for that matter, form a group. This group has as identity a
clean (solved) cube, its inverse is any solution algorithm, and its
operator is any cube twiddle. It has 18 generators: U, U2, U', D, D2,
D', L, L2, L', R, R2, R', F, F2, F' , B, B2 and B'. This is for the
group of the 3x3x3 cube. The total number of elements is approximately
4.3x10^(19) or 43  quintillion.
Hana a kostky

> Hana Bizek <hbizek@ameritech.net> wrote in message
> news:387DD6C1.24CC@ameritech.net...
> > Hello, all,
> > some of you  will know me from previous postings. My  quest for
> > 3-dimensional Rubik's cobe designers is still a dismal zero. There is a
> > real danger that I am the first to have come up with this. Please, if
> > you know of anyone who does  something similar to what I have done, let
> > me know. The page to look at is http://cube.misto.cz. Please go down to
> > the Appendices and click on "Thre-dimensional designs of Dr. Hana M.
> > Bizek." Thank you.
> >
> > Instead of providing me with a name of a fellow designer, the above page
> > ended up as a link to rather prestigious sites. Some are:
> > * Mensa CR at http://www.mensa.cz/cz/odkazy
> > * Sir Roger Penrose Society at http://welcome.to/SRPS
> > * The Geometry Junkyard (I forgot the URL, will look it up upon request)
> > * MSO Worldwide, Ltd at http://www.msoworld.com
> >
> > The last site, MSO Worldwide, has launched a new links directory. Our
> > site belongs to the Mental Skills - Mechanical Puzzles category. Please
> > have a look at it, and, if you think it's good, vote for our site. And
> > look around. It's a fascinating site for any and all puzzle lovers, and
> > math lovers.
> >
> > Cheers,
> >
> > Hana M. Bizek
> > hbizek@ameritech.net

Hana Bizek <hbizek@ameritech.net> wrote in message
news:38884C17.50BD@ameritech.net...
> Androcles wrote:
> >
> > Sorry Hana, no offense intended.  I just had not heard you mention it,
> > that's all. :-)
> > Androcles
>
> I am not offended. Amused is more like it.
>
> The reason I did not mention the fact is that I felt it to be too
> obvious. Besides, the original purpose of my post was not to give a
> lecture, but to ask you if you know of someone, who creates
> 3-dimensional designs from Rubik's cubes. But I devote a whole chapter
> in my book to thie fact that elements in a Rubik's cube form a
> mathematical group. (The book has been published, so it is not a
> manuscript any more. :-) ) If you can get a hold of it, you will find
> group information in Chapter 2.
>
> Permit me to disclose one tidbit about myself. I studied some group
> theory as part of my PhD dissertation in theoretical solid-state
> physics. So I am not a mathematician, but simewhat close. :-)
> Hana a kostky
Okay, well, for amusement only:
There is a well-known little puzzle known as the "Knight's Tour".
The object of the puzzle is to have the (chess) knight visit all 64 squares
on the chessboard
once and once only, according to his usual move, being 1 square in x, two
squares in y, or 1 square in y and two in x.
First question:
After the knight has completed the tour, what is the least number of moves
needed to return to the starting position?
Second question:
Can this result be modified by taking a different tour?
Suppose we have Rubik's cube, and suggest a tour such that all possible
outcomes of the various configurations are presented. We begin with the
identity.
Question  three, analogous to the knight's tour.
How many configurations are there, each to be visited once and once only?
Question four.
Is this possible?
Question five:
How many moves does it take to return to the identity?
Androcles

Sorry Hana, no offense intended.  I just had not heard you mention it,
that's all. :-)
Androcles

Androcles wrote:
> 
> Sorry Hana, no offense intended.  I just had not heard you mention it,
> that's all. :-)
> Androcles

I am not offended. Amused is more like it.

The reason I did not mention the fact is that I felt it to be too
obvious. Besides, the original purpose of my post was not to give a
lecture, but to ask you if you know of someone, who creates
3-dimensional designs from Rubik's cubes. But I devote a whole chapter
in my book to thie fact that elements in a Rubik's cube form a
mathematical group. (The book has been published, so it is not a
manuscript any more. :-) ) If you can get a hold of it, you will find
group information in Chapter 2.

Permit me to disclose one tidbit about myself. I studied some group
theory as part of my PhD dissertation in theoretical solid-state
physics. So I am not a mathematician, but simewhat close. :-)
Hana a kostky

Androcles wrote:
[snip]
> Okay, well, for amusement only:
> There is a well-known little puzzle known as the "Knight's Tour".
> The object of the puzzle is to have the (chess) knight visit all 64 squares
> on the chessboard
> once and once only, according to his usual move, being 1 square in x, two
> squares in y, or 1 square in y and two in x.
> First question:
> After the knight has completed the tour, what is the least number of moves
> needed to return to the starting position?
> Second question:
> Can this result be modified by taking a different tour?
[snip]

I am not sure I understand the first question (least number of moves),
but the Knight's tour puzzle has a known solution. You can find an
optimal tour by selecting always your next move according to the rule:
The available moves from your next move's spot has to be minimal. So, to
elaborate: If the Knight can move into positions x1, x2,..x8 (in
general), and denote by nm(xn) the number of available moves at place
xn, then a valid next move (that leads to a solution) would be xi, where
nm(xi)=min{nm(xj),j=1..8}.

An interesting side effect to a solution followed by the above
algorithm, is that if you number the squares according to the order
visited, you get a magic square. Since the chessboard is 8x8, this would
be a non-trivial magic square, contrary to odd sided squares which can
get formed by laying the natural numbers in diagonals.
Cheerio

> Androcles

-- 
Ioannis Galidakis jgal@ath.forthnet.gr
<http://www.crosswinds.net/athens/~jgal/main.html>
______________________________________
"Blessed are the ones who can laugh at themselves,
for they shall never cease to be amused."

On Fri, 21 Jan 2000 23:39:24 GMT, "Androcles" <androcles@home.com>
wrote:

>There is a well-known little puzzle known as the "Knight's Tour".
>The object of the puzzle is to have the (chess) knight visit all 64 squares
>on the chessboard
(snip)
>Suppose we have Rubik's cube, and suggest a tour such that all possible
>outcomes of the various configurations are presented. We begin with the
>identity.
>Question  three, analogous to the knight's tour.
>How many configurations are there, each to be visited once and once only?
>Question four.
>Is this possible?
>Question five:
>How many moves does it take to return to the identity?

There may be other ways to interpret *tour* in the context of a Rubik
cube but this one is a lot of fun.

Using a Rubiks cube simulator with macro capability such as
http://www.mud.ca/cube/cube.html
enter a sequence of moves.
How many repetitions of these identical moves will return the cube to
its original state?
What sequence  requires the most repetitions to restore?


John

John Bailey <jmb184@frontiernet.net> wrote in message
news:3889ab01.487646@news.frontiernet.net...
> On Fri, 21 Jan 2000 23:39:24 GMT, "Androcles" <androcles@home.com>
> wrote:
>
> >There is a well-known little puzzle known as the "Knight's Tour".
> >The object of the puzzle is to have the (chess) knight visit all 64
squares
> >on the chessboard
> (snip)
> >Suppose we have Rubik's cube, and suggest a tour such that all possible
> >outcomes of the various configurations are presented. We begin with the
> >identity.
> >Question  three, analogous to the knight's tour.
> >How many configurations are there, each to be visited once and once only?
> >Question four.
> >Is this possible?
> >Question five:
> >How many moves does it take to return to the identity?
>
> There may be other ways to interpret *tour* in the context of a Rubik
> cube but this one is a lot of fun.
>
> Using a Rubiks cube simulator with macro capability such as
> http://www.mud.ca/cube/cube.html
> enter a sequence of moves.
> How many repetitions of these identical moves will return the cube to
> its original state?
> What sequence  requires the most repetitions to restore?
John, that's quite a good simulator, but it didn't work in quite the way I
expected. I can rotate the entire cube using the right mouse button, but I
don't seem to be able to drag a slice as per the instructions. When I use
the keyboard, only the middle slice turns. What am I doing wrong?
I also note that the operation u1d1 returns the identity, as expected,
the operation u1u3 returns the identity, as expected, and u3 = d1,
but u1u1u1u1d1d1d1d1, while correctly returning the identity, actually goes
through the moves, whereas u1u3d1d3 does not.
Since u3 = d1, and u2=d2, the use of "d" becomes redundant.
While I applaud the programmer for his effort, I find this simulator rather
difficult to create a tour.
Androcles

Please disregard my previous post.
I misread the instructions for
http://www.mud.ca/cube/cube.html
Androcles

John Bailey wrote:
> 
> On Fri, 21 Jan 2000 23:39:24 GMT, "Androcles" <androcles@home.com>
> wrote:
> 
> >There is a well-known little puzzle known as the "Knight's Tour".
> >The object of the puzzle is to have the (chess) knight visit all 64 squares
> >on the chessboard
> (snip)
> >Suppose we have Rubik's cube, and suggest a tour such that all possible
> >outcomes of the various configurations are presented. We begin with the
> >identity.
> >Question  three, analogous to the knight's tour.
> >How many configurations are there, each to be visited once and once only?
> >Question four.
> >Is this possible?
> >Question five:
> >How many moves does it take to return to the identity?
> 
> There may be other ways to interpret *tour* in the context of a Rubik
> cube but this one is a lot of fun.
> 
> Using a Rubiks cube simulator with macro capability such as
> http://www.mud.ca/cube/cube.html
> enter a sequence of moves.
> How many repetitions of these identical moves will return the cube to
> its original state?
> What sequence  requires the most repetitions to restore?
> 
> John

I cannot answer this one, but any sequence that is an element of a
subgroup of the original Rubik's cube  group has a specuified cyclicity.
This distinguishes a sequence that may be used as part of asolution
algorithm brom random twiddling.I talk about this in my book.
Hana a kostky

Androcles wrote:
> 
> Hana Bizek <hbizek@ameritech.net> wrote in message
> news:38884C17.50BD@ameritech.net...
> > Androcles wrote:
> > >
> > > Sorry Hana, no offense intended.  I just had not heard you mention it,
> > > that's all. :-)
> > > Androcles
> >
> > I am not offended. Amused is more like it.
> >
> > The reason I did not mention the fact is that I felt it to be too
> > obvious. Besides, the original purpose of my post was not to give a
> > lecture, but to ask you if you know of someone, who creates
> > 3-dimensional designs from Rubik's cubes. But I devote a whole chapter
> > in my book to thie fact that elements in a Rubik's cube form a
> > mathematical group. (The book has been published, so it is not a
> > manuscript any more. :-) ) If you can get a hold of it, you will find
> > group information in Chapter 2.
> >
> > Permit me to disclose one tidbit about myself. I studied some group
> > theory as part of my PhD dissertation in theoretical solid-state
> > physics. So I am not a mathematician, but simewhat close. :-)
> > Hana a kostky
> Okay, well, for amusement only:
> There is a well-known little puzzle known as the "Knight's Tour".
> The object of the puzzle is to have the (chess) knight visit all 64 squares
> on the chessboard
> once and once only, according to his usual move, being 1 square in x, two
> squares in y, or 1 square in y and two in x.
> First question:
> After the knight has completed the tour, what is the least number of moves
> needed to return to the starting position?
> Second question:
> Can this result be modified by taking a different tour?
> Suppose we have Rubik's cube, and suggest a tour such that all possible
> outcomes of the various configurations are presented. We begin with the
> identity.
> Question  three, analogous to the knight's tour.
> How many configurations are there, each to be visited once and once only?
> Question four.
> Is this possible?
> Question five:
> How many moves does it take to return to the identity?
> Androcles

I am sorry to respond this late, Androcles, but I needed to think things
over. I am not familiar with the Knight's tour puzzle. Indeed, I have
some hazy notion of how a knight moves, but your explanation of this
point is good and I understand it.
Please look ar these remarks as coming from a relative ignoramus, but if
a tour is a continous parh across a chessboard, then you do not have
that in case of a Pubik's cube. One sequence may position a certain
cubie in ine place, anorther sequence may position it elsewhere on the
whole cube.It's an entirely different approachh, and one that is not
easily equated.
Hana a kostky

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Hana Bizek <hbizek@ameritech.net> wrote in message =
news:388AF552.61B0@ameritech.net...
> Androcles wrote:
> >=20
> > Hana Bizek <hbizek@ameritech.net> wrote in message
> > news:38884C17.50BD@ameritech.net...
> > > Androcles wrote:
> > > >
> > > > Sorry Hana, no offense intended.  I just had not heard you =
mention it,
> > > > that's all. :-)
> > > > Androcles
> > >
> > > I am not offended. Amused is more like it.
> > >
> > > The reason I did not mention the fact is that I felt it to be too
> > > obvious. Besides, the original purpose of my post was not to give =
a
> > > lecture, but to ask you if you know of someone, who creates
> > > 3-dimensional designs from Rubik's cubes. But I devote a whole =
chapter
> > > in my book to thie fact that elements in a Rubik's cube form a
> > > mathematical group. (The book has been published, so it is not a
> > > manuscript any more. :-) ) If you can get a hold of it, you will =
find
> > > group information in Chapter 2.
> > >
> > > Permit me to disclose one tidbit about myself. I studied some =
group
> > > theory as part of my PhD dissertation in theoretical solid-state
> > > physics. So I am not a mathematician, but simewhat close. :-)
> > > Hana a kostky
> > Okay, well, for amusement only:
> > There is a well-known little puzzle known as the "Knight's Tour".
> > The object of the puzzle is to have the (chess) knight visit all 64 =
squares
> > on the chessboard
> > once and once only, according to his usual move, being 1 square in =
x, two
> > squares in y, or 1 square in y and two in x.
> > First question:
> > After the knight has completed the tour, what is the least number of =
moves
> > needed to return to the starting position?
> > Second question:
> > Can this result be modified by taking a different tour?
> > Suppose we have Rubik's cube, and suggest a tour such that all =
possible
> > outcomes of the various configurations are presented. We begin with =
the
> > identity.
> > Question  three, analogous to the knight's tour.
> > How many configurations are there, each to be visited once and once =
only?
> > Question four.
> > Is this possible?
> > Question five:
> > How many moves does it take to return to the identity?
> > Androcles
>=20
> I am sorry to respond this late, Androcles, but I needed to think =
things
> over. I am not familiar with the Knight's tour puzzle. Indeed, I have
> some hazy notion of how a knight moves, but your explanation of this
> point is good and I understand it.
> Please look ar these remarks as coming from a relative ignoramus, but =
if
> a tour is a continous parh across a chessboard, then you do not have
> that in case of a Pubik's cube. One sequence may position a certain
> cubie in ine place, anorther sequence may position it elsewhere on the
> whole cube.It's an entirely different approachh, and one that is not
> easily equated.
> Hana a kostky
Actually, Hana, the knight's tour is really quite simple.
I was using it as an example. Beginning at the corner, 0, the operator =
(2x1y) or (1x2y) places the knight at either of the squares marked 1 and =
63. Because the board is symmetric, these are actually equivalent
(view with a fixed font)
+---+---+---+---+---+---+---+---+
| 0 |   | 2 |   |   |   |   |   |
+---+---+---+---+---+---+---+---+
|   |   | 63| 2 |   |   |   |   |=20
+---+---+---+---+---+---+---+---+
|   | 1 |   |   |   |   |   |   |=20
+---+---+---+---+---+---+---+---+
|   |   |   | 2 |   |   |   |   |=20
+---+---+---+---+---+---+---+---+
| 2 |   | 2 |   |   |   |   |   |=20
+---+---+---+---+---+---+---+---+
|   |   |   |   |   |   |   |   |=20
+---+---+---+---+---+---+---+---+
|   |   |   |   |   |   |   |   |=20
+---+---+---+---+---+---+---+---+
|   |   |   |   |   |   |   |   |
+---+---+---+---+---+---+---+---+

From there, the choices are the squares marked 2 - a return to 0 isn't =
permitted for a tour. The very last move must be from square 63 to 0.
Now it seems to me that the rotation operators of Rubik's cube could be =
laid out in a string. There are 8 corner (3-color) cubies and 8 edge =
cubies (2-color) that can occupy any given face, and of course the =
middle of the face doesn't ever change (we are not considering a =
rotation of the entire cube in the view of the observer).=20
However, once a corner cubie has been selected for position on one face, =
it inevitably determines the corner color on two other faces. Hence the =
last face is predetermined by the color selection for 5 faces.
In the simulator
http://www.mud.ca/cube/cube.html
John Bailey found, rotation of the center slice is actually a bit of a =
cheat, since it is really two moves, one each for the outer slices and a =
rotation of the entire cube in the view of the observer. In the tour, we =
would consider this an invalid move. I do applaud the simulator, though =
- it is a very nice piece of work.=20
So, keeping the orientation of the face centers constant with respect to =
the viewer, the question becomes this. Can the six operations (UDLRFB) =
be performed in such a sequence that all members of the group are found =
once and once only, the last move returning to the identity?
Your intuition here, it would seem, is that it cannot.
My intuition is that it can, just like the knight's tour.
Now can you (or any other mathematician here) prove such a sequence =
exists?
Assuming it can, then we have completely solved Rubik's cube for any =
given random scramble, from identity through all possibilities and back =
to identity, albeit by exhaustion.
The question then becomes:
Given that from any given scramble there is a path back to the identity =
described by the tour and by the inverse operations that caused the =
scramble to begin with, which of these two routes is the shorter path? =
(In other words, did the scramble exceed a point halfway through the =
tour?)
The next question then becomes - is there a shortcut from any given =
point on the tour that returns us to the identity with fewer moves than =
the inverse operations or completion of the tour?
I see the tour as a circle, and a shortcut would be a chord on that =
circle.=20
My intuition tells me there are such chords, but I cannot prove it. Such =
a chord would be the equivalent of finding the shortest route home (the =
identity) from any square on the knight's tour, and they obviously =
exist.
Androcles








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<DIV><BR><FONT face=3D"Courier New" size=3D2>Hana Bizek =
&lt;hbizek@ameritech.net&gt;=20
wrote in message news:388AF552.61B0@ameritech.net...<BR>&gt; Androcles=20
wrote:<BR>&gt; &gt; <BR>&gt; &gt; Hana Bizek =
&lt;hbizek@ameritech.net&gt; wrote=20
in message<BR>&gt; &gt; news:38884C17.50BD@ameritech.net...<BR>&gt; &gt; =
&gt;=20
Androcles wrote:<BR>&gt; &gt; &gt; &gt;<BR>&gt; &gt; &gt; &gt; Sorry =
Hana, no=20
offense intended.&nbsp; I just had not heard you mention it,<BR>&gt; =
&gt; &gt;=20
&gt; that's all. :-)<BR>&gt; &gt; &gt; &gt; Androcles<BR>&gt; &gt; =
&gt;<BR>&gt;=20
&gt; &gt; I am not offended. Amused is more like it.<BR>&gt; &gt; =
&gt;<BR>&gt;=20
&gt; &gt; The reason I did not mention the fact is that I felt it to be=20
too<BR>&gt; &gt; &gt; obvious. Besides, the original purpose of my post =
was not=20
to give a<BR>&gt; &gt; &gt; lecture, but to ask you if you know of =
someone, who=20
creates<BR>&gt; &gt; &gt; 3-dimensional designs from Rubik's cubes. But =
I devote=20
a whole chapter<BR>&gt; &gt; &gt; in my book to thie fact that elements =
in a=20
Rubik's cube form a<BR>&gt; &gt; &gt; mathematical group. (The book has =
been=20
published, so it is not a<BR>&gt; &gt; &gt; manuscript any more. :-) ) =
If you=20
can get a hold of it, you will find<BR>&gt; &gt; &gt; group information =
in=20
Chapter 2.<BR>&gt; &gt; &gt;<BR>&gt; &gt; &gt; Permit me to disclose one =
tidbit=20
about myself. I studied some group<BR>&gt; &gt; &gt; theory as part of =
my PhD=20
dissertation in theoretical solid-state<BR>&gt; &gt; &gt; physics. So I =
am not a=20
mathematician, but simewhat close. :-)<BR>&gt; &gt; &gt; Hana a =
kostky<BR>&gt;=20
&gt; Okay, well, for amusement only:<BR>&gt; &gt; There is a well-known =
little=20
puzzle known as the "Knight's Tour".<BR>&gt; &gt; The object of the =
puzzle is to=20
have the (chess) knight visit all 64 squares<BR>&gt; &gt; on the=20
chessboard<BR>&gt; &gt; once and once only, according to his usual move, =
being 1=20
square in x, two<BR>&gt; &gt; squares in y, or 1 square in y and two in=20
x.<BR>&gt; &gt; First question:<BR>&gt; &gt; After the knight has =
completed the=20
tour, what is the least number of moves<BR>&gt; &gt; needed to return to =
the=20
starting position?<BR>&gt; &gt; Second question:<BR>&gt; &gt; Can this =
result be=20
modified by taking a different tour?<BR>&gt; &gt; Suppose we have =
Rubik's cube,=20
and suggest a tour such that all possible<BR>&gt; &gt; outcomes of the =
various=20
configurations are presented. We begin with the<BR>&gt; &gt; =
identity.<BR>&gt;=20
&gt; Question&nbsp; three, analogous to the knight's tour.<BR>&gt; &gt; =
How many=20
configurations are there, each to be visited once and once only?<BR>&gt; =
&gt;=20
Question four.<BR>&gt; &gt; Is this possible?<BR>&gt; &gt; Question=20
five:<BR>&gt; &gt; How many moves does it take to return to the=20
identity?<BR>&gt; &gt; Androcles<BR>&gt; <BR>&gt; I am sorry to respond =
this=20
late, Androcles, but I needed to think things<BR>&gt; over. I am not =
familiar=20
with the Knight's tour puzzle. Indeed, I have<BR>&gt; some hazy notion =
of how a=20
knight moves, but your explanation of this<BR>&gt; point is good and I=20
understand it.<BR>&gt; Please look ar these remarks as coming from a =
relative=20
ignoramus, but if<BR>&gt; a tour is a continous parh across a =
chessboard, then=20
you do not have<BR>&gt; that in case of a Pubik's cube. One sequence may =

position a certain<BR>&gt; cubie in ine place, anorther sequence may =
position it=20
elsewhere on the<BR>&gt; whole cube.It's an entirely different =
approachh, and=20
one that is not<BR>&gt; easily equated.<BR>&gt; Hana a =
kostky<BR>Actually, Hana,=20
the knight's tour is really quite simple.</FONT></DIV>
<DIV><FONT face=3D"Courier New" size=3D2>I was using it as an example. =
Beginning at=20
the corner, 0, the operator (2x1y) or (1x2y) places the knight at either =
of the=20
squares marked 1 and 63. Because the board is symmetric, these are =
actually=20
equivalent</FONT></DIV>
<DIV><FONT face=3D"Courier New" size=3D2>(view with a fixed =
font)</FONT></DIV>
<DIV><FONT face=3D"Courier New" size=3D2>
<DIV><FONT face=3D"Courier New" size=3D2><FONT face=3D"Courier New" =
size=3D2>
<DIV><FONT face=3D"Courier New" size=3D2>+---+<FONT face=3D"Courier New" =

size=3D2>---+<FONT face=3D"Courier New" size=3D2>--<FONT face=3D"Courier =
New"=20
size=3D2>-+---+<FONT face=3D"Courier New" size=3D2>---+<FONT =
face=3D"Courier New"=20
size=3D2>---+<FONT face=3D"Courier New" size=3D2>--<FONT face=3D"Courier =
New"=20
size=3D2>-+---+</FONT></FONT></FONT></FONT></FONT></FONT></FONT></FONT></=
DIV>
<DIV>| 0 |&nbsp;  |&nbsp;2 |&nbsp; &nbsp;|&nbsp;&nbsp; |&nbsp;&nbsp;=20
|&nbsp;&nbsp; |&nbsp;&nbsp; |</DIV>
<DIV><FONT face=3D"Courier New" size=3D2>
<DIV><FONT face=3D"Courier New" size=3D2><FONT face=3D"Courier New" =
size=3D2><FONT=20
face=3D"Courier New" size=3D2>+---+<FONT face=3D"Courier New" =
size=3D2>---+<FONT=20
face=3D"Courier New" size=3D2>--<FONT face=3D"Courier New" =
size=3D2>-+---+<FONT=20
face=3D"Courier New" size=3D2>---+<FONT face=3D"Courier New" =
size=3D2>---+<FONT=20
face=3D"Courier New" size=3D2>--<FONT face=3D"Courier New"=20
size=3D2>-+---+</FONT></FONT></FONT></FONT></FONT></FONT></FONT></FONT>
<DIV>|&nbsp;&nbsp;&nbsp;|&nbsp; =
&nbsp;|&nbsp;63|&nbsp;2&nbsp;|&nbsp;&nbsp;=20
|&nbsp;&nbsp; |&nbsp;&nbsp; |&nbsp;&nbsp; |<FONT face=3D"Courier New" =
size=3D2>=20
<DIV><FONT face=3D"Courier New" size=3D2>+---+<FONT face=3D"Courier New" =

size=3D2>---+<FONT face=3D"Courier New" size=3D2>--<FONT face=3D"Courier =
New"=20
size=3D2>-+---+<FONT face=3D"Courier New" size=3D2>---+<FONT =
face=3D"Courier New"=20
size=3D2>---+<FONT face=3D"Courier New" size=3D2>--<FONT face=3D"Courier =
New"=20
size=3D2>-+---+</FONT></FONT></FONT></FONT></FONT></FONT></FONT></FONT></=
DIV>
<DIV>|&nbsp;  |&nbsp;1&nbsp;|&nbsp;  |&nbsp; &nbsp;|&nbsp;&nbsp; =
|&nbsp;&nbsp;=20
|&nbsp;&nbsp; |&nbsp;&nbsp; |<FONT face=3D"Courier New" size=3D2>=20
<DIV><FONT face=3D"Courier New" size=3D2>+---+<FONT face=3D"Courier New" =

size=3D2>---+<FONT face=3D"Courier New" size=3D2>--<FONT face=3D"Courier =
New"=20
size=3D2>-+---+<FONT face=3D"Courier New" size=3D2>---+<FONT =
face=3D"Courier New"=20
size=3D2>---+<FONT face=3D"Courier New" size=3D2>--<FONT face=3D"Courier =
New"=20
size=3D2>-+---+</FONT></FONT></FONT></FONT></FONT></FONT></FONT></FONT></=
DIV>
<DIV>|&nbsp;&nbsp;&nbsp;|&nbsp;&nbsp;&nbsp;|&nbsp;  =
|&nbsp;2&nbsp;|&nbsp;&nbsp;=20
|&nbsp;&nbsp; |&nbsp;&nbsp; |&nbsp;&nbsp; |<FONT face=3D"Courier New" =
size=3D2>=20
<DIV><FONT face=3D"Courier New" size=3D2>+---+<FONT face=3D"Courier New" =

size=3D2>---+<FONT face=3D"Courier New" size=3D2>--<FONT face=3D"Courier =
New"=20
size=3D2>-+---+<FONT face=3D"Courier New" size=3D2>---+<FONT =
face=3D"Courier New"=20
size=3D2>---+<FONT face=3D"Courier New" size=3D2>--<FONT face=3D"Courier =
New"=20
size=3D2>-+---+</FONT></FONT></FONT></FONT></FONT></FONT></FONT></FONT></=
DIV>
<DIV>|&nbsp;2&nbsp;|&nbsp;&nbsp;&nbsp;|&nbsp;2 =
|&nbsp;&nbsp;&nbsp;|&nbsp;&nbsp;=20
|&nbsp;&nbsp; |&nbsp;&nbsp; |&nbsp;&nbsp; |<FONT face=3D"Courier New" =
size=3D2>=20
<DIV><FONT face=3D"Courier New" size=3D2>+---+<FONT face=3D"Courier New" =

size=3D2>---+<FONT face=3D"Courier New" size=3D2>--<FONT face=3D"Courier =
New"=20
size=3D2>-+---+<FONT face=3D"Courier New" size=3D2>---+<FONT =
face=3D"Courier New"=20
size=3D2>---+<FONT face=3D"Courier New" size=3D2>--<FONT face=3D"Courier =
New"=20
size=3D2>-+---+</FONT></FONT></FONT></FONT></FONT></FONT></FONT></FONT></=
DIV>
<DIV>|&nbsp;&nbsp;&nbsp;|&nbsp;&nbsp;&nbsp;|&nbsp;&nbsp;=20
|&nbsp;&nbsp;&nbsp;|&nbsp;&nbsp; |&nbsp;&nbsp; |&nbsp;&nbsp; =
|&nbsp;&nbsp;=20
|<FONT face=3D"Courier New" size=3D2>=20
<DIV><FONT face=3D"Courier New" size=3D2>+---+<FONT face=3D"Courier New" =

size=3D2>---+<FONT face=3D"Courier New" size=3D2>--<FONT face=3D"Courier =
New"=20
size=3D2>-+---+<FONT face=3D"Courier New" size=3D2>---+<FONT =
face=3D"Courier New"=20
size=3D2>---+<FONT face=3D"Courier New" size=3D2>--<FONT face=3D"Courier =
New"=20
size=3D2>-+---+</FONT></FONT></FONT></FONT></FONT></FONT></FONT></FONT></=
DIV>
<DIV>|&nbsp;&nbsp;&nbsp;|&nbsp;&nbsp;&nbsp;|&nbsp;&nbsp;=20
|&nbsp;&nbsp;&nbsp;|&nbsp;&nbsp; |&nbsp;&nbsp; |&nbsp;&nbsp; =
|&nbsp;&nbsp;=20
|<FONT face=3D"Courier New" size=3D2>=20
<DIV><FONT face=3D"Courier New" size=3D2>+---+<FONT face=3D"Courier New" =

size=3D2>---+<FONT face=3D"Courier New" size=3D2>--<FONT face=3D"Courier =
New"=20
size=3D2>-+---+<FONT face=3D"Courier New" size=3D2>---+<FONT =
face=3D"Courier New"=20
size=3D2>---+<FONT face=3D"Courier New" size=3D2>--<FONT face=3D"Courier =
New"=20
size=3D2>-+---+</FONT></FONT></FONT></FONT></FONT></FONT></FONT></FONT></=
DIV>
<DIV>|&nbsp;&nbsp;&nbsp;|&nbsp;&nbsp;&nbsp;|&nbsp;&nbsp;=20
|&nbsp;&nbsp;&nbsp;|&nbsp;&nbsp; |&nbsp;&nbsp; |&nbsp;&nbsp; =
|&nbsp;&nbsp;=20
|</DIV>
<DIV>
<DIV><FONT face=3D"Courier New" size=3D2>+---+<FONT face=3D"Courier New" =

size=3D2>---+<FONT face=3D"Courier New" size=3D2>--<FONT face=3D"Courier =
New"=20
size=3D2>-+---+<FONT face=3D"Courier New" size=3D2>---+<FONT =
face=3D"Courier New"=20
size=3D2>---+<FONT face=3D"Courier New" size=3D2>--<FONT face=3D"Courier =
New"=20
size=3D2>-+---+</FONT></FONT></FONT></FONT></FONT></FONT></FONT></FONT></=
DIV></DIV></FONT></DIV></FONT></DIV></FONT></DIV></FONT></DIV></FONT></DI=
V></FONT></DIV></FONT></FONT></DIV></FONT></DIV></FONT></FONT></DIV>
<DIV>&nbsp;</DIV>
<DIV>From there, the choices are the squares marked 2 - a return to 0 =
isn't=20
permitted for a tour. The very last move&nbsp;must be from square 63 to =
0.</DIV>
<DIV>Now it seems to me that the rotation operators of Rubik's cube =
could be=20
laid out in a string. There are 8 corner (3-color) cubies and 8 edge =
cubies=20
(2-color) that can occupy any given face, and of course the middle of =
the face=20
doesn't ever change (we are not considering a rotation of the entire =
cube in the=20
view of the observer). </DIV>
<DIV>However, once a corner cubie has been selected for position on one =
face,=20
it&nbsp;inevitably determines the corner color on two other faces. Hence =
the=20
last face is predetermined by the color selection for 5 faces.</DIV>
<DIV>In the simulator</DIV>
<DIV><A=20
href=3D"http://www.mud.ca/cube/cube.html">http://www.mud.ca/cube/cube.htm=
l</A><BR>John=20
Bailey found, rotation of the center slice is actually a bit of a cheat, =
since=20
it is really two moves, one each for the outer slices and a rotation of =
the=20
entire cube in the view of the observer. In the tour, we would consider =
this an=20
invalid move. I do applaud the&nbsp;simulator, though -&nbsp;it is a =
very nice=20
piece of work. </DIV>
<DIV>So, keeping the orientation of the face centers constant with =
respect to=20
the viewer, the question becomes this. Can the six operations (UDLRFB) =
be=20
performed in such a sequence that all members of the group are found =
once and=20
once only, the last move returning to the identity?</DIV>
<DIV>Your intuition here, it would seem, is that it cannot.</DIV>
<DIV>My intuition is that it can, just like the knight's tour.</DIV>
<DIV>Now can you (or any other mathematician here) prove&nbsp;such a =
sequence=20
exists?</DIV>
<DIV>Assuming it can, then we have completely solved Rubik's cube for =
any given=20
random scramble, from identity through all possibilities and back to =
identity,=20
albeit by exhaustion.</DIV>
<DIV>The question then becomes:</DIV>
<DIV>Given that from any given scramble there is a path back to the =
identity=20
described by the tour and by the inverse operations that caused the =
scramble to=20
begin with, which of these two routes is the shorter path? (In other =
words, did=20
the scramble exceed a point halfway through the tour?)</DIV>
<DIV>The next question then becomes - is there a shortcut from any given =
point=20
on the tour that returns us to the identity with fewer moves than the =
inverse=20
operations or completion of the tour?</DIV>
<DIV>I see the tour as a circle, and a shortcut would be a chord on that =
circle.=20
</DIV>
<DIV>My intuition tells me there are&nbsp;such chords, but I cannot =
prove it.=20
Such a chord&nbsp;would be the equivalent of finding the shortest route =
home=20
(the identity) from any square on the knight's tour, and they obviously=20
exist.</DIV>
<DIV>Androcles</DIV>
<DIV>&nbsp;</DIV>
<DIV>&nbsp;</DIV>
<DIV>&nbsp;</DIV>
<DIV>&nbsp;</DIV>
<DIV>&nbsp;</DIV>
<DIV>&nbsp;</DIV>
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On Sat, 22 Jan 2000 16:06:18 GMT, "Androcles" <androcles@home.com>
wrote:

>
>John Bailey <jmb184@frontiernet.net> wrote in message
>news:3889ab01.487646@news.frontiernet.net...
>> On Fri, 21 Jan 2000 23:39:24 GMT, "Androcles" <androcles@home.com>
>> wrote:
>> (snip)
>> >Suppose we have Rubik's cube, and suggest a tour such that all possible
>> >outcomes of the various configurations are presented. 
>> There may be other ways to interpret *tour* in the context of a Rubik
>> cube but this one is a lot of fun.
>> http://www.mud.ca/cube/cube.html
>John, that's quite a good simulator, but it didn't work in quite the way I
>expected. 
> When I use
>the keyboard, only the middle slice turns. What am I doing wrong?
CAPITALS select the outer slices, lower case the inner slices.
The instructions also might help with the other problems you
mentioned.  (See, I did not say: RTFM)
>While I applaud the programmer for his effort, I find this simulator rather
>difficult to create a tour.
He undoubtably did not intend it for exploring group theory or
hamiltonian tours.

John